Problem: Determine how many solutions exist for the system of equations. ${-8x-2y = -2}$ ${y = 6x-3}$
Answer: Convert both equations to slope-intercept form: ${-8x-2y = -2}$ $-8x{+8x} - 2y = -2{+8x}$ $-2y = -2+8x$ $y = 1-4x$ ${y = -4x+1}$ ${y = 6x-3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x+1}$ ${y = 6x-3}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.